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Inscribed Right Triangle Theorem

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter. Conversely, if one side of an inscribed triangle's sides is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.
Circle circumscribed to a right triangle

These statements are commonly referred to as Thales' Theorem and the converse of Thales' Theorem.


Thales' Theorem

Consider a right triangle with inscribed in a circle.

According to the Inscribed Angle Theorem, the measure of is half the measure of its intercepted arc,

Substituting into the equation above gives that Then, is a semicircle, implying that the hypotenuse of is a diameter of the circle.

Converse of Thales' Theorem

Consider a triangle inscribed in a circle such that one side of the triangle is a diameter of the circle.

Since is a diameter, then is a semicircle and then Now, the Inscribed Angle Theorem gives a relation between this arc and the angle opposite to the diameter.
This implies that is a right angle, which makes a right triangle.